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In the book "Foundations of Set Theory" by Fraenkel, Bar-Hillel and Levy (second edition, 1973), on page 153, there is the following statement about Ackermann set theory:

"By means of the axiom of foundation one can prove the existence of the class $P(V)$ which consists of all subclasses of $V$." ($V$ is the class of all sets).

My question is:

Is there a proper class $X$ in Ackermann set theory, for which $P(X)$ does not exist in Ackermann set theory?

($P(X)$ is the collection of all subclasses of $X$.)

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